test(cron): own-profile bot-chat turn from a multiplexed tick runs in the ticking home

Regression guard for #119858. The fix itself is already on main: 3b0fe0cc2b pinned
the child HERMES_HOME to the override-aware source home and 786c0e3f9d moved the
spawn onto served_profile_child_env(target_home=home, inherit_credentials=True).
No existing test asserted the OWN-profile (no -p) leg A->B->A under multiplex; these
two do (red at 786c0e3f9dc~1, green on main).

tests/cron/test_bot_chat_own_profile_multiplex_home.py

@@ -0,0 +1,73 @@
+"""``deliver: bot-chat`` (the job's OWN profile, no ``-p``) from a multiplexed tick runs the delivery
+turn in the ticking profile's home, never the gateway's launch home (#119858).
+
+The multiplexed ticker binds the served profile as a ``HERMES_HOME`` override that never reaches
+``os.environ``; the child env must be derived from the override-aware home (``served_profile_child_env``),
+never from ``os.environ.copy()``.
+"""
+from __future__ import annotations
+
+import subprocess
+
+import pytest
+
+import cron.scheduler_delivery as delivery
+from agent.secret_scope import set_multiplex_active
+from hermes_constants import reset_hermes_home_override, set_hermes_home_override
+
+
+@pytest.fixture
+def multiplexer(tmp_path, monkeypatch):
+    """A default-hosted multiplexer whose launch ``.env`` is in ``os.environ``, serving alpha and beta."""
+    root = tmp_path / "hermes"
+    for name in ("alpha", "beta"):
+        home = root / "profiles" / name
+        home.mkdir(parents=True)
+        (home / ".env").write_text(f"ANTHROPIC_API_KEY=sk-{name}\n{name.upper()}_ONLY=1\n", encoding="utf-8")
+    (root / ".env").write_text("ANTHROPIC_API_KEY=sk-root\nLAUNCH_ONLY_MARKER=1\n", encoding="utf-8")
+    monkeypatch.setenv("HERMES_HOME", str(root))
+    monkeypatch.setenv("ANTHROPIC_API_KEY", "sk-root")
+    monkeypatch.setenv("LAUNCH_ONLY_MARKER", "1")
+    captured: dict = {}
+
+    def _fake_turn(argv, env, report_path, timeout):
+        captured.update(env=dict(env), argv=list(argv))
+        return subprocess.CompletedProcess(argv, 0, stdout="", stderr="")
+
+    monkeypatch.setattr(delivery, "_run_bot_chat_turn", _fake_turn)
+    set_multiplex_active(True)
+    try:
+        yield root, captured
+    finally:
+        set_multiplex_active(False)
+
+
+def test_an_own_profile_bot_chat_turn_runs_in_the_ticking_profiles_home(multiplexer):
+    """A→B→A: each served profile's bare ``bot-chat`` child carries THAT profile's home and
+    credentials; the launch profile's marker is absent; no ``-p`` is synthesised."""
+    root, captured = multiplexer
+    for name in ("alpha", "beta", "alpha"):
+        home = root / "profiles" / name
+        token = set_hermes_home_override(str(home))
+        try:
+            assert delivery._deliver_to_bot_chat({"id": f"j-{name}", "name": "nightly"}, "the brief", "") is None
+        finally:
+            reset_hermes_home_override(token)
+        env = captured["env"]
+        assert env["HERMES_HOME"] == str(home)
+        assert env["ANTHROPIC_API_KEY"] == f"sk-{name}"
+        assert "LAUNCH_ONLY_MARKER" not in env, "the launch profile's env reached a served profile's turn"
+        assert "-p" not in captured["argv"]
+
+
+def test_an_explicit_target_profile_still_wins_over_the_ticking_profile(multiplexer):
+    """Control: ``bot-chat:beta`` fired from alpha's tick is built for beta, not for alpha."""
+    root, captured = multiplexer
+    token = set_hermes_home_override(str(root / "profiles" / "alpha"))
+    try:
+        assert delivery._deliver_to_bot_chat({"id": "j", "name": "nightly"}, "the brief", "beta") is None
+    finally:
+        reset_hermes_home_override(token)
+    assert captured["env"]["HERMES_HOME"] == str(root / "profiles" / "beta")
+    assert captured["env"]["ANTHROPIC_API_KEY"] == "sk-beta"
+    assert "ALPHA_ONLY" not in captured["env"]